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at some point I was wondering what about the researcher publishing their paper in an expensive journal and it turns out that the author does not make a lot of money from that, so it really is about fucking this exploitative system and not doing any substantial harm to the researcher, if anyone was worrying like I was

This is about Sci-Hub. yeah we get it.. gatekeep knowledge and protect the interests of capital…

Pick a point inside a triangle and drop perpendicular projections onto the sides. These define another triangle. Repeat, with the same point but within the new triangle. Do the same thing once more. The fourth triangle now has the same angles as the first one, although it’s much smaller and it’s rotated.

Inflating regular pentagon through underlying stars. The sequence of side lengths is essentially Fibonacci.

13 X 2022

I dedicated the weekend to meeting with people from the machine learning club, helping my friend through her analysis homework and studying category theory for one of my subjects. then I did mostly the complex analysis homework

here are some wannabe aesthetic notes

my main goal at the time was to truly understand yoneda's lemma and the main intuition I have is that sometimes we shouldn't study the category C, but thw category of all functors from C to Set

after studying for a few hours I can say that the concept became a bit more intuitive

one of the problems in my "putnam homework" was to calculate the product of all differences of distinct n-th roots of unity – or so I thought. for a few days I believed that my solution doesn't work. I ended up with a disgusting fomula interating cosines of obscure angles but the visual intuition is neat, especially for an odd n. aaand that's no surprise since it turns out I'm fucking illiterate. not distinct roots, just differences of distinct roots, so that the whole thing is symmetric and there is no distinction of n odd vs n even

anyway I finally solved it, so that's nice!

I completed 5 out of 10 problems, which was my goal, so I should stop now and do my commutative algebra homework. there is one more exercise I want to solve:

the complex polynomial P with integer coefficients is such that |P(z)| ≤ 2 ∀z∈S¹. how many non-zero coefficients can P have?

I'm almost there with it and it's really cool

ofc the opportunity to include pretty drawings in my homework couldn't be wasted

during my category theory tutorial the professor asked me to show my solution on the blackboard. I was kinda stressed because now is the first time when I have my lectures and tutorials in english and on top of that this is a grad course. that whole morning I was fighting to stay awake, after the blackboard incident I didn't have to anymore

this is what I did

this week is likely to be the hardest out of many proceeding ones, because I won't have the weekend for studying (it's my grandma's birthday) so I need to use the maximum of my time during the week and get as much done as possible. I still need to do two homeworks, and study the theory. I am trying to learn how to prioritize and plan things, this is still a huge problem for me

I found an interesting youtube channel: Justin Sung. he talks about how to study/ how to learn and I like what he says, because it just makes so much sense. it's been a while since I started suspecting that methods such as flash cards or simple note-taking don't work and his content explains very well why they indeed might not work. it's very inspiring to see a professional confirm one's intuition

I've been thinking about how different math feels after three years of consistently doing it. it's a sad thought, because I used to get super excited about learning new things and solving problems, whereas now my standards seem to be higher..?

I spent the day doing exercises from galois theory and statistics, in preparation for the tests I have soon. it felt like a chore. sure, the exercises were easy and uninteresting, I decided to start from the basics, so there is that. however, in general practicing like this became a routine and there used to be a sense of mystery around it that is now gone

when I don't have any deadlines but feel like doing some math the obvious choice is to learn something that will be useful in the future. more homological algebra, algebraic geometry, K-theory, or digging deeper into the topics I already am familiar with. all of those are good candidates and I used to be very motivated to just learn something new. but here comes to paradox of choice, where every option is good, but there isn't a great one

I think I might be annoyed with always learning the prerequisites for something not yet defined. it did feel exciting when I was studying the modules of tangles so that I could answer an open question, it doesn't feel as exciting to learn about the galois theory to pass a test. a metaphor comes to mind. doing math without a fulfilling goal feels like taking a walk – it's rather nice, I enjoy going on walks. with a fulfilling goal it feels like walking towards a destination such that the walk itself is a pleasant activity, but I really want to get to said destination. by that I mean that I still enjoy simply learning new stuff and working on exercises, but it doesn't feel as fulfilling as it used to, how much walking without getting anywhere can you do in three years? you can do the same thing in prison

three years is nothing compared to how much knowledge and experience is necessary to do actual research, I know that. I fail to feel it, but I know it. when I am asking myself what state of mind is the most fulfilling I'd say exploration, discovery, getting an idea that is new to me and seemingly comes from nowhere, not just an obvious corollary of what I've seen in lectures, an insight, an act of creating. I suppose all those things are to be found in the future, but god how long do I have to wait

on a more pragmatic and realistic note, I think I'll talk to my professors about what I can do to speed up that process. I'll ask them how the actual research feels and how they went from being a student learning basic concepts to where they are now

a question to those of you who are more experienced than me: does this even sound familiar at all? what were you like as a student and what took you to where you are now? how does math feel after 3, 5, 10 years?

people using a matrix as just a bunch of numbers in a grid or a way to summarize some elaborate calculation instead of a way of notating a linear transformation (or at least a set of points) feels kind of genuinely profane to me. like its one of the only times i feel like i "get" the concept of the profane. how could you do that to her

My favourite fucked up math fact™ is the Sharkovskii theorem:

For any continuous function f: [a,b] -> [a,b], if there exists a periodic point of order 3 (i.e. f(f(f(x))) = x for some x in [a,b] and not f(x) = x or f²(x) = x), then there exists a periodic point of ANY order n.¹

Yes you read that right. If you can find a point of order 3 then you can be sure that there is a point of order 4, 5, or even 142857 in your interval. The assumption is so innocent but I cannot understate how ridiculous the result is.²

For a (relatively) self-contained proof, see this document (this downloads a pdf).

(footnotes under read more)

¹ The interval does not have to be closed, but it should be connected. (a,b), (a,b] and [a,b) all work.

² Technically the result is even stronger! The natural numbers admit a certain ordering called the Sharkovskii ordering which starts with the odd primes 3 > 5 > 7 > ... , then doubles of primes, then quadruples of primes and so forth until you get no more primes left, ending the ordering in 2³ > 2² > 2. Sharkovskii's theorem actually says that if you have a periodic point of order k, then you have periodic points of any order less than k in the Sharkovskii ordering. It is frankly ridiculous how somehow prime numbers make their way into this mess.

from now I’ve got to start chasing regular discomfort, because otherwise comfort isn’t comfort but stagnation. you can’t have one without the other. to be challenged is to grow and it’s the only way to actually feel at peace

1 X 2022

new month huh

yesterday the commutative algebra teacher sent out the first homework assignment. you know, fuck the holiday, we need that grind

I have a week to solve it but I started yesterday as I was so excited

we need to prove some elementary properties of commutative unitary rings and I am enjoying it, I completed a half of the exercises so far. I can tell that the intuition acquired from studying module theory is paying off. many of the requested properties are the special cases of what I encountered during my module venture, so I feel like I understand them quite well. the problem I come across is how to write it down in a rigorous way, but I guess this is why we're supposed to do those exercises

I just got home from the math camp, it was so exhausting. I am not used to being around people all the time, so I my tolerance for interactions is low. I'm glad I went there tho, because I gained some teaching experience – my lecture, choosing contest problems and then grading the solutions

my university offers jobs as graders, older students can make some extra money checking homeworks of younger ones. the requirement is to have a decent GPA, which I don't have so I'm afraid they won't accept me. I don't know how decent exactly tho, so I'm going to try. in particular I might get bonus points for my extracurricular activities, giving talks at conferences and the grading I did at the camp. I'm so done with being poor, I hope I get in. otherwise I might start looking for some programming jobs, not for this academic year but in general, to find out what I could do at all

a few days ago I found a book that I wish I had found sooner: Vector Analysis, Klaus Janich

these are some of the chapters I needed a few months ago for my analysis course. the book is written like a novel and contains many interesting examples. on the bright side there are chapters about riemannian manifolds and other stuff that I haven't yet had an opportunity to study, so I plan to skim through the topics I already know and stay longer at those new to me

well, the sememster starts on tuesday so I don't have much time for that book, but as a sidequest it seems just right

We need books that affect us like a disaster, that grieve us deeply, like being banished into forests far from everyone. A book must be the axe for the frozen sea within us. That is my belief.

Franz Kafka